Experience shows that the propagation of sound waves is generally characterized by a low damping within the fluid in which they propagate.

We therefore neglect the dissipative processes (thermal conductivity and viscosity), which amounts to applying the isentropic character of the propagation of sound waves and thus to assume a perfect fluid.

The only forces considered are the pressure forces (gravity is neglected).

That is \(\mu_0\), \(P_0\) and \(T_0\) the characteristics of the fluid at rest (assumed uniform), we note :

• \(\mu =\mu_T-\mu_0\), the variation of density of the fluid (\(\left |\mu\right | <<\mu_T,\mu_0\) )

• \(p=P-P_0\), the variation of fluid pressure, also called acoustic pressure (\(\left |p \right |<<P,P_0\))

• \(\vec v\) the vector velocity of a fluid particle (no rest)

The acoustic approximation is to consider that the magnitudes \(\vec v\), \(\mu\) and \(p\) are infinitesimal of the same order (and their spatial and temporal derivatives also).

In particular, the calculations will be made to order \(1\) in this infinitely small.

Magnitude of the acoustic pressure :

Overpressure likely to be detected by the ear typically range from \(100 \;Pa\) (his painfull) to \(10^{-5}\; Pa\) (hearing threshold), covering \(7\) decades.

The propagation of sound in a fluid can be studied by considering that the fluid makes small movements isentropic.

As part of the acoustic approximation, the coefficient of isentropic compressibility gives :

\(\chi _S = - \frac{1}{V}\left( {\frac{{\partial V}}{{\partial P}}} \right)_S = \frac{1}{{\mu _T }}\left( {\frac{{\partial \mu _T }}{{\partial P}}} \right)_S \approx \frac{1}{{\mu _0 }}\frac{\mu }{p}\;\;\;\;\;\;\;\;\;\;so\;\;\;\;\;\;\;\;\;\;\mu \approx \mu _0 \chi _S p\)

• The mass conservation equation is :

  \(div(\mu _T \vec v) + \frac{{\partial \mu _T }}{{\partial t}} = 0\)

  Either, with \(\mu =\mu_T-\mu_0\) :

  \(div((\mu _0 + \mu )\vec v) + \frac{{\partial \mu }}{{\partial t}} = 0\)

  Whence :

  \(\mu _0 div(\vec v) + \mu div(\vec v) + \vec v.\overrightarrow {grad} (\mu) + \frac{{\partial \mu }}{{\partial t}} = 0\)

  Linear approximation (or acoustic) : we limit the following \(1st\) order terms.

  Therefore :

  \(\mu _0 div(\vec v) + \frac{{\partial \mu }}{{\partial t}} = 0\)

• The equation of motion of the fluid is here Euler's equation (no viscosity) :

  \(\mu _T \left( {\frac{{\partial \vec v}}{{\partial t}} + (\vec v.\overrightarrow {grad} )\vec v} \right) = - \overrightarrow {grad} P + \vec f_v\)

  The static volume force (eg, \(\vec f_v =\mu_T \vec g\)) is compensated by the gradient of static pressure \(P_0\).

  The influence of variations (for example \(\mu \vec g\)) is in practice negligible compared to the pressure gradient \(p\).

  The equation of motion becomes, after linearization :

  \(\mu _0 \frac{{\partial \vec v}}{{\partial t}} = - \overrightarrow {grad} p\)

• More we recall the relationship between acoustic pressure and the change in density :

  \(\mu \approx \mu _0 \chi _S p\)

• Eliminating variable \(\mu\), one obtains the coupled equations system :

  \(\frac{{\partial \vec v}}{{\partial t}} = - \frac{1}{{\mu _0 }}\overrightarrow {grad} p\;\;\;\;\;and\;\;\;\;\;\frac{{\partial p}}{{\partial t}} = - \frac{1}{{\chi _S }}div(\vec v)\)

• For overpressure \(p\) :

  Calculating the divergence of the previous left equation :

  \(div\left( {\frac{{\partial \vec v}}{{\partial t}}} \right) = - \frac{1}{{\mu _0 }}div\left( {\overrightarrow {grad} p} \right)\)

  Is :

  \(\frac{\partial }{{\partial t}}\left( { - \chi _S \frac{{\partial p}}{{\partial t}}} \right) = - \frac{1}{{\mu _0 }}\Delta p\;\;\;\;\;\;\;so\;\;\;\;\;\;\;\Delta p - \mu _0 \chi _S \frac{{\partial ^2 p}}{{\partial t^2 }} = 0\)

  We recognize the d'Alembert wave equation ; the speed of the sound waves is deduced :

  \(\;\Delta p - \frac{1}{{c_s^2 }}\frac{{\partial ^2 p}}{{\partial t^2 }} = 0\;\;\;\;\;\;\;\;\;\;with\;\;\;\;\;\;\;\;\;\;c_s = \frac{1}{{\sqrt {\mu _0 \chi _S } }}\)

• For speed \(\vec v\) :

  Writing that :

  \(div(\vec v) = - \chi _S \frac{{\partial p}}{{\partial t}}\)

  By taking the gradient while deriving the equation :

  \(\mu _0 \frac{{\partial \vec v}}{{\partial t}} = - \overrightarrow {grad} p\)

  With respect to time :

  \(- \overrightarrow {grad} \left( {\frac{{\partial p}}{{\partial t}}} \right) = \frac{1}{{\chi _S }}\overrightarrow {grad} (div(\vec v)) = \frac{1}{{\chi _S }}\left( {\overrightarrow {rot} (\overrightarrow {rot}( \vec v)) + \Delta \vec v} \right) = \mu _0 \frac{{\partial ^2 \vec v}}{{\partial t^2 }}\)

  We have used the relation : (see the lesson Vector calculus)

  \(\overrightarrow {rot} (\overrightarrow {rot} (\vec A)) = \overrightarrow {grad} (div(\vec A) )- \Delta \vec A\)

  Very often, there will be the velocity field of a form \(\vec v=v(x,t)\;\vec u_x\), Therefore \(\vec {rot}(\vec v)=\vec 0\). This result can be generalized.

  The equation satisfied by the velocity field becomes :

  \(\;\Delta \vec v- \frac{1}{{c_s^2 }}\frac{{\partial ^2 \vec v}}{{\partial t^2 }} = 0\;\;\;\;\;\;\;\;\;\;with\;\;\;\;\;\;\;\;\;\;c_s = \frac{1}{{\sqrt {\mu _0 \chi _S } }}\)

  We find an identical d'Alembert relation to that obtained for the pressure.